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Fermi problems

November 10, 2012

We often know more about quantities than we give ourselves credit for.

For example, we may say that we don’t how many people were at a gathering. But if asked whether there were 20 or 200, we answer “Perhaps 50”. We can even guess large crowd sizes with acceptable accuracy – we know whether there are closer to 10000 or 50000 people at a sports event. Then again, if someone asks me how many grains of sand make up a reasonable size beach, I wouldn’t know where to stop writing zeroes – but could I make a reasonable guess within, say, 10 minutes?

This is an example of some mathematical or scientific problems, for which we also have only a rough knowledge of the numbers involved. Nevertheless, we can come up with estimates accurate enough to give us a useful feel for the problem.

These are called Fermi problems.

In thinking about these problems, it is important to state (especially to yourself) your assumptions and estimates. Feel free to allow for a factor of 10, 100, 1000 or more either way, depending on the scale of the problem.

Scientific notation

In discussing Fermi Problems, very large numbers are often needed, for which we generally use ‘scientific notation‘ e.g.

  • 30,000 = 3×104 (i.e. 3 followed by 4 zeroes)
  • 400,000,000,000 =4×1011 (i.e. 4 followed by 11 zeroes)
  • 10,000,000,000,000,000,000,000,000,000,000,000,000 = 1×1037 or just 1037 (i.e. 1 followed by 37 zeroes)

Simply put, write the first digit, followed by 10 to the power of the number of zeroes. So, some typical Fermi problems…

1. Words

a. How many words did you speak yesterday?

I know different people will give different answers to this one – certainly the case in our household! You’ll have to decide when a word is a word, mumbled, murmured, gargled or grunted though it may be. Still, for most of us over 5 years old, this should make only a marginal difference.

 b. How many words will you have spoken in your entire life?


c. How many words have been spoken – ever?

Double ditto.

Useful facts:
– People speak at an average of about 150 words per minute. You may adjust this up or down.
– There have been something like 73 (or 76) billion people ever born.

– A modern western average life expectancy is about 80 years, and an historical one is, say, 35 years.

2. Raindrops

a. How many raindrops fell on your locality during the last wet day?

We are after an order of magnitude estimate here.

b. How many raindrops have ever fallen anywhere on Earth?

Another more extreme Fermi problem– I think a factor of 1000 or so either way is as much as we could expect. The somewhat surprising thing is to compare your estimate with that for the previous problem.

c. How many raindrops has each molecule of water been part of?

Assume that, over a long enough time, all water on Earth is well mixed. Those water molecules sure get around !

Useful facts:
– Raindrops vary between 0.1mm and 9mm, but are typically, say, 1mm in diameter.
– World average annual rainfall is 990mm i.e. very close to 1m.
– World surface area is 510,072,200km2 or about 5×108km.2 = 5×1014m2.
– Global water volume is 1,386,000,000 cubic kilometers (km3) i.e. about 109 km3 = 1018 m3.

3. Leaves on trees

a. How many leaves are on the largest tree near where you live?

Assuming it’s not mid-winter 🙂 We are after an order of magnitude estimate here – feel free to allow for a factor of, say, 10 either way. You should be able to find a tree with more than 100,000 = 105 leaves.

b. How many leaves fall each year on Earth?

That’s a lot to sweep up !

Useful facts:
– A very nice discussion of leaves on a tree is here.
– There are about
400 billion = 4×1011 trees on earth (but going down!)

4. Space dust

About 40 tons of material falls from space each day. Assume that this is a reasonable average over time for this space dust.

a. Because of this material, how much larger is the Earth’s diameter since, say, the Cambrian era (540 million years ago)?

1mm, 1cm, 1m, 1km, 100km? I had no idea at all of the answer to this before working it out.

b. How does the mass of a large meteorite compare with the normal rain of dust, say over the 65 million years since that event?

The Cretaceous–Paleogene boundary was caused by a meteorite about 10km in diameter. This impact led to the demise of the dinosaurs.

Useful facts:
– The earth’s diameter is 6371km.
– The average density of earth’s material is about 5.515 g/cm3 (same link as above).
– Let’s use the same density for space dust, once it is incorporated into the earth’s material.

You can make up your own problems

Some suggestions:
– How many animals are alive now? How many have ever lived? Let’s restrict ourselves here to those bigger than, say, 1cm long.
– How long would you need to live in a forest to be within earshot of a large tree falling over – naturally – at the end of its life?
– How many tears have been cried?

  1. I had a go at the “raindrops fallen since the Earth began” one in my summary of your article on my own blog: and came up with 4.2E33. Would you mind checking through my working? I’m a little rusty on my maths after 2 years away from A level maths!

    • Hi Jake,

      Yes, they look pretty good to me.

      Regards Mark

    • Hi Jake,

      Something I forgot to add is that I think knowing the likely (in)accuracy of your answer is just as important as the answer itself here.

      So I think the .2 is definitely unjustified, and even the 4 is very doubtful. With such large number and such uncertain data to work with, it is more sensible to think of the uncertainty in the exponent. I suspect the allowable error of the exponent to be at least 5%, suggesting a factor of at least 30 either way – a range of about 10^(33.5+-1.5) i.e. between 10^32 and 10^35.

      regards mark

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